A mention of the original tune Woody used comes in this recent thread, with the words "Woody performed the song as at least one of the Peoples' Artists hoots in New York, but his original tune was almost a flat monotone and it (the tune) was not well received."

I'd like to hear it sometime - I can think of some great songs which people have put down as having pretty well no tune. You don't necessarily need to use a lot of notes.